One-dimensional Problem of an Elastic Bar

We consider the problem of an elastic bar where the elastic modulus is inhomogeneous. The bar is subjected to a state of uniaxial force F at the end point. The equation of equilibrium for the bar is given by 

As shown in Fig. 7.1, we consider a one-dimensional bar with a periodic structure at the microscale level. 

Ichikawa and A.P.S. Selvadurai, Transport Phenomena in Porous Media, 

We introduce a macroscale coordinate system and a microscale one for the general system x as 

The periodic microstructiu e is referred to as a unit cell. In homogenization analysis we assume that the size of the unit cell is sufficiently small, and by taking the limit 0 we derive a system of differential equations that relates the microscale behavior to the microscale characteristics. 

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